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Recovery Rates, Default Probabilitiesand the Credit Cycle
Max Bruche and Carlos Gonzalez-Aguado
CEMFI
BIS Stress Testing workshop, March 2007
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Motivation
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ate
3050
1985 1990 1995 2000 2005
avg.
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over
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te
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
The research questions & method
The questions:
• Can you exploit the joint time-series behaviour of default rates andrecovery rates to characterize and forecast credit risk?
• How bad is it to treat recovery rates as constant (or independent ofdefault probabilities)?
The method:
• We propose an econometric model in which both the time variation indefault probabilities and recovery rate distributions is driven by anunobserved Markov chain, the “credit cycle”.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Related literature
• Default probabilities or rating transitions vary over time, and arerelated to macro variables.(Bangia et al. (2002), Nickell et al. (2000))
• Recovery rates and default probabilities are contemporaneouslynegatively related.(Altman et al. 2006, Acharya et al. 2007)
• Recovery rates and default probabilities can be modelled as functionsof observed covariates.(Chava et al. 2006)
• Theory:
• Recovery rates should be related to the state of the industry: Shleiferand Vishny (1992).
• RBC and credit: Bernanke and Gertler (1989), Williamson (1987)
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Summary of results
Observed covariates versus latent factor:
• The proposed model describes the data well, and does better thanmany models based on observed covariates.
• E.g. out-of-sample rolling RMSE for predicted recovery rates is22.86%. (Chava et al. 2006: 24.96%)
We can use the estimated model to look at what happens when we gofrom constant to time-varying recovery rate distributions. We get
• higher estimates of tail risk,(for a sample portfolio, the 99% VaR goes from 2.6% to 2.9%.)
• the same expected losses,
• bigger swings in spreads over the cycle. Average spreads over thecycle are not affected.
Caveats?
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Summary of results
Observed covariates versus latent factor:
• The proposed model describes the data well, and does better thanmany models based on observed covariates.
• E.g. out-of-sample rolling RMSE for predicted recovery rates is22.86%. (Chava et al. 2006: 24.96%)
We can use the estimated model to look at what happens when we gofrom constant to time-varying recovery rate distributions. We get
• higher estimates of tail risk,(for a sample portfolio, the 99% VaR goes from 2.6% to 2.9%.)
• the same expected losses,
• bigger swings in spreads over the cycle. Average spreads over thecycle are not affected.
Caveats?
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Summary of results
Observed covariates versus latent factor:
• The proposed model describes the data well, and does better thanmany models based on observed covariates.
• E.g. out-of-sample rolling RMSE for predicted recovery rates is22.86%. (Chava et al. 2006: 24.96%)
We can use the estimated model to look at what happens when we gofrom constant to time-varying recovery rate distributions. We get
• higher estimates of tail risk,(for a sample portfolio, the 99% VaR goes from 2.6% to 2.9%.)
• the same expected losses,
• bigger swings in spreads over the cycle. Average spreads over thecycle are not affected.
Caveats?
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Summary of results
Observed covariates versus latent factor:
• The proposed model describes the data well, and does better thanmany models based on observed covariates.
• E.g. out-of-sample rolling RMSE for predicted recovery rates is22.86%. (Chava et al. 2006: 24.96%)
We can use the estimated model to look at what happens when we gofrom constant to time-varying recovery rate distributions. We get
• higher estimates of tail risk,(for a sample portfolio, the 99% VaR goes from 2.6% to 2.9%.)
• the same expected losses,
• bigger swings in spreads over the cycle. Average spreads over thecycle are not affected.
Caveats?
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
The basic idea
The DGP works as follows:
• The state of the credit cycle is determined by a two-state Markovchain.
• The number of defaulting firms is drawn using the state-dependentdefault probability.
• For each defaulting firm, we draw a recovery rate from thestate-dependent recovery rate distribution.
Dependence:
• Conditional on the state, defaults are independent, recoveries betweenfirms are independent, and the number of defaulting firms andrecoveries are independent.
• As a consequence, (unconditional) dependence is driven entirely bythe (unobserved) state of the credit cycle.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
The basic idea
The DGP works as follows:
• The state of the credit cycle is determined by a two-state Markovchain.
• The number of defaulting firms is drawn using the state-dependentdefault probability.
• For each defaulting firm, we draw a recovery rate from thestate-dependent recovery rate distribution.
Dependence:
• Conditional on the state, defaults are independent, recoveries betweenfirms are independent, and the number of defaulting firms andrecoveries are independent.
• As a consequence, (unconditional) dependence is driven entirely bythe (unobserved) state of the credit cycle.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Specific assumptions about functional forms
• Conditional on the state of the cycle, default arrival is described bydiscrete hazards of the form
λt = (1 + exp {γ0 + γ1ct + γ2Xt})−1 .
(t: time, ct: cycle, Xt: economy-wide variables)
• Recoveries for each default event are drawn from a beta distribution.
• The parameters of this beta distribution are given by:
αtis = exp {δ0 + δ1ct + · · ·+ δ6Xt} (1)
βtis = exp {ζ0 + ζ1ct + · · ·+ ζ6Xt} (2)
(i: firm, s: seniority)
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Specific assumptions about functional forms
• Conditional on the state of the cycle, default arrival is described bydiscrete hazards of the form
λt = (1 + exp {γ0 + γ1ct + γ2Xt})−1 .
(t: time, ct: cycle, Xt: economy-wide variables)
• Recoveries for each default event are drawn from a beta distribution.
• The parameters of this beta distribution are given by:
αtis = exp {δ0 + δ1ct + · · ·+ δ6Xt} (1)
βtis = exp {ζ0 + ζ1ct + · · ·+ ζ6Xt} (2)
(i: firm, s: seniority)
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Specific assumptions about functional forms
• Conditional on the state of the cycle, default arrival is described bydiscrete hazards of the form
λt = (1 + exp {γ0 + γ1ct + γ2Xt})−1 .
(t: time, ct: cycle, Xt: economy-wide variables)
• Recoveries for each default event are drawn from a beta distribution.
• The parameters of this beta distribution are given by:
αtis = exp {δ0 + δ1ct + · · ·+ δ6Xt} (1)
βtis = exp {ζ0 + ζ1ct + · · ·+ ζ6Xt} (2)
(i: firm, s: seniority)
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Specific assumptions about functional forms
• Conditional on the state of the cycle, default arrival is described bydiscrete hazards of the form
λt = (1 + exp {γ0 + γ1ct + γ2Xt})−1 .
(t: time, ct: cycle, Xt: economy-wide variables)
• Recoveries for each default event are drawn from a beta distribution.
• The parameters of this beta distribution are given by:
αtis = exp {δ0 + δ1ct + · · ·+ δ6Xt} (1)
βtis = exp {ζ0 + ζ1ct + · · ·+ ζ6Xt} (2)
(i: firm, s: seniority)
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Estimation
• The model can be easily be estimated using a version of the Hamiltonfilter (MLE).
• For this we need the number of defaulting firms, and non-defaultingfirms in each period, and a recovery rate for each default event.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Data sources
We use data from two main sources,
• the Altman-NYU Salomon Center Corporate Bond Default MasterDatabase, consisting of prices just after default of more than 2,000bonds of US firms from 1974 to 2005, with issuers and dates, and a“bond category”,
• and the annual default rates reported in Standard & Poor’s QuarterlyDefault Update from May 2006.
Assuming that both the Altman data and Standard & Poor’s data exhibitthe same default rates, we can obtain the number of non-defaulting firmsin each year.
• We augment this with GDP growth, investment growth,unemployment, the S&P 500 index, the VIX, the slope of the termstructure, and an NBER recession indicator.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Data sources
We use data from two main sources,
• the Altman-NYU Salomon Center Corporate Bond Default MasterDatabase, consisting of prices just after default of more than 2,000bonds of US firms from 1974 to 2005, with issuers and dates, and a“bond category”,
• and the annual default rates reported in Standard & Poor’s QuarterlyDefault Update from May 2006.
Assuming that both the Altman data and Standard & Poor’s data exhibitthe same default rates, we can obtain the number of non-defaulting firmsin each year.
• We augment this with GDP growth, investment growth,unemployment, the S&P 500 index, the VIX, the slope of the termstructure, and an NBER recession indicator.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Data sources
We use data from two main sources,
• the Altman-NYU Salomon Center Corporate Bond Default MasterDatabase, consisting of prices just after default of more than 2,000bonds of US firms from 1974 to 2005, with issuers and dates, and a“bond category”,
• and the annual default rates reported in Standard & Poor’s QuarterlyDefault Update from May 2006.
Assuming that both the Altman data and Standard & Poor’s data exhibitthe same default rates, we can obtain the number of non-defaulting firmsin each year.
• We augment this with GDP growth, investment growth,unemployment, the S&P 500 index, the VIX, the slope of the termstructure, and an NBER recession indicator.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Data sources
We use data from two main sources,
• the Altman-NYU Salomon Center Corporate Bond Default MasterDatabase, consisting of prices just after default of more than 2,000bonds of US firms from 1974 to 2005, with issuers and dates, and a“bond category”,
• and the annual default rates reported in Standard & Poor’s QuarterlyDefault Update from May 2006.
Assuming that both the Altman data and Standard & Poor’s data exhibitthe same default rates, we can obtain the number of non-defaulting firmsin each year.
• We augment this with GDP growth, investment growth,unemployment, the S&P 500 index, the VIX, the slope of the termstructure, and an NBER recession indicator.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Data sources
We use data from two main sources,
• the Altman-NYU Salomon Center Corporate Bond Default MasterDatabase, consisting of prices just after default of more than 2,000bonds of US firms from 1974 to 2005, with issuers and dates, and a“bond category”,
• and the annual default rates reported in Standard & Poor’s QuarterlyDefault Update from May 2006.
Assuming that both the Altman data and Standard & Poor’s data exhibitthe same default rates, we can obtain the number of non-defaulting firmsin each year.
• We augment this with GDP growth, investment growth,unemployment, the S&P 500 index, the VIX, the slope of the termstructure, and an NBER recession indicator.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Macro variables versus the business cycle
Model 1 Model 2 Model 3 Model 4Explanatory variables for λ (default probability)
constant constant constant constantcycle log GDP growth cycle
log GDP growthExplanatory variables for α,β (recovery rates)
constant constant constant constantcycle log GDP growth cycle
log GDP growthsenioritya senioritya senioritya senioritya
AIC -37.73 -405.31 -128.96 -412.79BIC 0.0727 -0.1549 0.0188 -0.1290
=⇒ Macro variables are significant, but do not contribute much to thefit.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Macro variables versus the business cycle
Model 1 Model 2 Model 3 Model 4Explanatory variables for λ (default probability)
constant constant constant constantcycle log GDP growth cycle
log GDP growthExplanatory variables for α,β (recovery rates)
constant constant constant constantcycle log GDP growth cycle
log GDP growthsenioritya senioritya senioritya senioritya
AIC -37.73 -405.31 -128.96 -412.79BIC 0.0727 -0.1549 0.0188 -0.1290
=⇒ Macro variables are significant, but do not contribute much to thefit.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Is the business cycle = credit cycle?
1985 1990 1995 2000 2005
0.0
0.2
0.4
0.6
0.8
1.0
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• The estimated credit downturns start earlier than NBER recessions,and end later.
• We investigate lead-lag relationships between macro variables andcredit variables and find that recovery rates Granger cause log GDPgrowth (very significant!).
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (1)
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (1)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (1)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (1)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (1)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (1)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (2)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (2)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (3)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
VaR Simulation (3)
0.00 0.01 0.02 0.03 0.04
020
4060
8010
012
014
0
Loss fraction
Den
sity
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Expected Loss
Expected loss is not affected
• SupposePD|E = 2% L|E = 30%PD|R = 10% L|R = 70%
P (E) = 1/2
=⇒ E[L] = 50%, E[PD] = 6%.Therefore
E[L · PD] = .5× 30%× 0.02 + .5× 70%× 0.1 = 3.8%E[L] · E[PD] = 3%
=⇒ E[L · PD]− E[L] · E[PD] = 80bp
• But E[L · PD]− E[L]E[PD] = Cov(L,PD).• In our data, Cov(avg. L, dfr) = 5bp.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Expected Loss
Expected loss is not affected
• SupposePD|E = 2% L|E = 30%PD|R = 10% L|R = 70%
P (E) = 1/2 =⇒ E[L] = 50%, E[PD] = 6%.
Therefore
E[L · PD] = .5× 30%× 0.02 + .5× 70%× 0.1 = 3.8%E[L] · E[PD] = 3%
=⇒ E[L · PD]− E[L] · E[PD] = 80bp
• But E[L · PD]− E[L]E[PD] = Cov(L,PD).• In our data, Cov(avg. L, dfr) = 5bp.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Expected Loss
Expected loss is not affected
• SupposePD|E = 2% L|E = 30%PD|R = 10% L|R = 70%
P (E) = 1/2 =⇒ E[L] = 50%, E[PD] = 6%.Therefore
E[L · PD] = .5× 30%× 0.02 + .5× 70%× 0.1 = 3.8%E[L] · E[PD] = 3%
=⇒ E[L · PD]− E[L] · E[PD] = 80bp
• But E[L · PD]− E[L]E[PD] = Cov(L,PD).• In our data, Cov(avg. L, dfr) = 5bp.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Expected Loss
Expected loss is not affected
• SupposePD|E = 2% L|E = 30%PD|R = 10% L|R = 70%
P (E) = 1/2 =⇒ E[L] = 50%, E[PD] = 6%.Therefore
E[L · PD] = .5× 30%× 0.02 + .5× 70%× 0.1 = 3.8%E[L] · E[PD] = 3%
=⇒ E[L · PD]− E[L] · E[PD] = 80bp
• But E[L · PD]− E[L]E[PD] = Cov(L,PD).
• In our data, Cov(avg. L, dfr) = 5bp.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Expected Loss
Expected loss is not affected
• SupposePD|E = 2% L|E = 30%PD|R = 10% L|R = 70%
P (E) = 1/2 =⇒ E[L] = 50%, E[PD] = 6%.Therefore
E[L · PD] = .5× 30%× 0.02 + .5× 70%× 0.1 = 3.8%E[L] · E[PD] = 3%
=⇒ E[L · PD]− E[L] · E[PD] = 80bp
• But E[L · PD]− E[L]E[PD] = Cov(L,PD).• In our data, Cov(avg. L, dfr) = 5bp.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle
Introduction The Model Data Some estimation results Implications for risk management Conclusion
Some conclusions
• We propose an econometric model in which default rates and recoveryrates are driven by an unobserved Markov chain.
• This describes the data well, and does better than many modelsbased on observed covariates.
We can use the estimated model to look at what happens when we gofrom constant to time-varying recovery rate distributions. We get
• higher estimates of tail risk,
• the same expected losses,
• bigger swings in spreads over the cycle. Average spreads over thecycle are not affected.
Max Bruche and Carlos Gonzalez-Aguado (CEMFI) Recovery Rates, Default Probabilities and the Credit Cycle